Stratified and Cluster Sampling -...
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Stratified and Cluster SamplingLecture 8
Sections 2.6, 2.8
Robb T. Koether
Hampden-Sydney College
Tue, Sep 9, 2008
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Outline
1 Homework Review
2 Introduction
3 Stratified Random SamplesExample
4 Estimating Parameters
5 Cluster SamplesExample
6 Stratified vs. Cluster
7 Assignment
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Homework Review
Exercise 1.17(a, b), p. 107In the past, about 10% of customers were dissatisfied withthe service received from local automobile dealers. Thisyear, the manufacturer has reason to believe that thepercentage of dissatisfied customers has decreased. Hedecides to test this theory.(a) State the competing hypotheses H0 and H1 clearly.(b) Out of a total of 40,000 customers, he decides to select
a simple random sample of size 120. Use yourcalculator with a seed value of 32 to give the labels ofthe first four selected customers.
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Homework Review
Solution (a, b)(a) H0: The percentage of dissatisfied customers is the
same as in the past.H1: The percentage of dissatisfied customers is lessthan in the past.
(b) The population size is N = 40, 000 and the sample sizeis n = 120. So we must choose 120 random numbersfrom 1 to 40,000. First enter 32→rand to set the seed.Then enter randInt(1,40000) in the TI-83 and pressENTER 5 times. The numbers
34318, 15553, 8461, 614, 38964
appear.
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Homework Review
Exercise 1.17(c, d), p. 107(c) The data were found to be not statistically significant at
α = 0.10 level. What type of error could have beenmade?
(d) Give a possible p-value for this test.
Solution (c, d)(c) “Not statistically significant” means that the null
hypothesis was accepted. Therefore, it is possible that aType II error was made.
(d) Since the null hypothesis was accepted, the p-valuemust have been greater than α. So a possible p-value is0.11. Another possible value is 0.12.
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Introduction
Suppose we know that our population is 60% male and40% female.Wouldn’t it make sense to ensure that our sample isalso 60% male and 40% female.After all, our goal is representativeness, notrandomness, right?If our sample size were n = 100, we could
Select a simple random sample of 60 males.Select a simple random sample of 40 females.
Is this a good idea?If we left it to chance, we might end up with 50 malesand 50 females.
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Introduction
Suppose we know that our population is 60% male and40% female.Wouldn’t it make sense to ensure that our sample isalso 60% male and 40% female.After all, our goal is representativeness, notrandomness, right?If our sample size were n = 100, we could
Select a simple random sample of 60 males.Select a simple random sample of 40 females.
Is this a good idea?If we left it to chance, we might end up with 50 malesand 50 females.
![Page 8: Stratified and Cluster Sampling - people.hsc.edupeople.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-09-11 · Stratified and Cluster Sampling Robb T. Koether Homework](https://reader033.fdocuments.us/reader033/viewer/2022050216/5f6260b6e4a09f591e48c158/html5/thumbnails/8.jpg)
Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Introduction
Suppose we know that our population is 60% male and40% female.Wouldn’t it make sense to ensure that our sample isalso 60% male and 40% female.After all, our goal is representativeness, notrandomness, right?If our sample size were n = 100, we could
Select a simple random sample of 60 males.Select a simple random sample of 40 females.
Is this a good idea?If we left it to chance, we might end up with 50 malesand 50 females.
![Page 9: Stratified and Cluster Sampling - people.hsc.edupeople.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-09-11 · Stratified and Cluster Sampling Robb T. Koether Homework](https://reader033.fdocuments.us/reader033/viewer/2022050216/5f6260b6e4a09f591e48c158/html5/thumbnails/9.jpg)
Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Introduction
Suppose we know that our population is 60% male and40% female.Wouldn’t it make sense to ensure that our sample isalso 60% male and 40% female.After all, our goal is representativeness, notrandomness, right?If our sample size were n = 100, we could
Select a simple random sample of 60 males.Select a simple random sample of 40 females.
Is this a good idea?If we left it to chance, we might end up with 50 malesand 50 females.
![Page 10: Stratified and Cluster Sampling - people.hsc.edupeople.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-09-11 · Stratified and Cluster Sampling Robb T. Koether Homework](https://reader033.fdocuments.us/reader033/viewer/2022050216/5f6260b6e4a09f591e48c158/html5/thumbnails/10.jpg)
Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Introduction
Suppose we know that our population is 60% male and40% female.Wouldn’t it make sense to ensure that our sample isalso 60% male and 40% female.After all, our goal is representativeness, notrandomness, right?If our sample size were n = 100, we could
Select a simple random sample of 60 males.Select a simple random sample of 40 females.
Is this a good idea?If we left it to chance, we might end up with 50 malesand 50 females.
![Page 11: Stratified and Cluster Sampling - people.hsc.edupeople.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-09-11 · Stratified and Cluster Sampling Robb T. Koether Homework](https://reader033.fdocuments.us/reader033/viewer/2022050216/5f6260b6e4a09f591e48c158/html5/thumbnails/11.jpg)
Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Introduction
Suppose we know that our population is 60% male and40% female.Wouldn’t it make sense to ensure that our sample isalso 60% male and 40% female.After all, our goal is representativeness, notrandomness, right?If our sample size were n = 100, we could
Select a simple random sample of 60 males.Select a simple random sample of 40 females.
Is this a good idea?If we left it to chance, we might end up with 50 malesand 50 females.
![Page 12: Stratified and Cluster Sampling - people.hsc.edupeople.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-09-11 · Stratified and Cluster Sampling Robb T. Koether Homework](https://reader033.fdocuments.us/reader033/viewer/2022050216/5f6260b6e4a09f591e48c158/html5/thumbnails/12.jpg)
Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Introduction
Suppose we know that our population is 60% male and40% female.Wouldn’t it make sense to ensure that our sample isalso 60% male and 40% female.After all, our goal is representativeness, notrandomness, right?If our sample size were n = 100, we could
Select a simple random sample of 60 males.Select a simple random sample of 40 females.
Is this a good idea?If we left it to chance, we might end up with 50 malesand 50 females.
![Page 13: Stratified and Cluster Sampling - people.hsc.edupeople.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-09-11 · Stratified and Cluster Sampling Robb T. Koether Homework](https://reader033.fdocuments.us/reader033/viewer/2022050216/5f6260b6e4a09f591e48c158/html5/thumbnails/13.jpg)
Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Introduction
Suppose we know that our population is 60% male and40% female.Wouldn’t it make sense to ensure that our sample isalso 60% male and 40% female.After all, our goal is representativeness, notrandomness, right?If our sample size were n = 100, we could
Select a simple random sample of 60 males.Select a simple random sample of 40 females.
Is this a good idea?If we left it to chance, we might end up with 50 malesand 50 females.
![Page 14: Stratified and Cluster Sampling - people.hsc.edupeople.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-09-11 · Stratified and Cluster Sampling Robb T. Koether Homework](https://reader033.fdocuments.us/reader033/viewer/2022050216/5f6260b6e4a09f591e48c158/html5/thumbnails/14.jpg)
Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Stratified Random Sample
Definition (Homogeneous)
A group is homogeneous if its member all have similarcharacteristics with regard to the variables of interest.
Definition (Stratum)A stratum is a homogeneous subset of the population.
Definition (Stratified random sampling)
Stratified random sampling is a sampling method in whichthe population is first divided into strata. Then a simplerandom sample is taken from each stratum. The resultsconstitute the sample.
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Examples
Possible strata:Male and female strata.Resident and non-resident strata.White, black, Hispanic, and Asian strata.Protestant, Catholic, Jewish, Muslim, etc., strata.
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Stratified Random Sampling
The population
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Stratified Random Sampling
The population
The strata
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Stratified Random Sampling
The population
The strata
Random sample from this stratum
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Stratified Random Sampling
The population
The strata
Random sample from this stratum
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Stratified Random Sampling
The population
Random samples from all strata
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Stratified Random Sampling
The stratified sample
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Example
Example (Stratified random sample)Let the population consist of males Andy, Bob, Charlie,Don, Ed, Fred, Greg, and Hank and females Pam,Queeny, Rachel, Susie, Terri, Ursula, Valerie, andWendy.Choose a stratified sample of size n = 8, where thestrata are the two sexes.Is the sample representative with regard to sex?
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Estimating Parameters
Suppose thatThe population is 60% male and 40% female.Our sample is 50% male and 50% female.Our variable has an average value of 10 for the malesand 15 for the females.
What is our best estimate for the variable’s average forthe population?
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Estimating Parameters
We need to compute a weighted average.
average = (0.60)(10) + (0.40)(15)= 6 + 6
= 12
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Cluster Sampling
Definition (Heterogeneous)
A group is heterogeneous if is members vary in regard tothe variables of interest in the same way that the populationvaries.
Definition (Cluster)A cluster is a heterogeneous subset of the population.
Definition (Cluster random sampling)
Cluster random sampling is a sampling method in which thepopulation is first divided into clusters. Then a simplerandom sample of clusters is taken. All the members of theselected clusters together constitute the sample.
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Cluster Sampling
Note that it is the clusters that are selected at random,not the individuals.It is hoped that each cluster by itself is representative ofthe population, i.e., each cluster is heterogeneous.
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Cluster Random Sampling
The population
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Cluster Random Sampling
The population
The clusters
![Page 29: Stratified and Cluster Sampling - people.hsc.edupeople.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-09-11 · Stratified and Cluster Sampling Robb T. Koether Homework](https://reader033.fdocuments.us/reader033/viewer/2022050216/5f6260b6e4a09f591e48c158/html5/thumbnails/29.jpg)
Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Cluster Random Sampling
The population
The clusters
Randomly select some clusters
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Cluster Random Sampling
The population
The clusters
Selected all member in the clusters
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Cluster Random Sampling
The cluster sample
![Page 32: Stratified and Cluster Sampling - people.hsc.edupeople.hsc.edu/faculty-staff/robbk/Math121/Lectures/Fall... · 2008-09-11 · Stratified and Cluster Sampling Robb T. Koether Homework](https://reader033.fdocuments.us/reader033/viewer/2022050216/5f6260b6e4a09f591e48c158/html5/thumbnails/32.jpg)
Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Example
Example (Cluster sample)Now suppose that
Andy, Bob, Pam, and Queeny live in Fredericksburg.Charlie, Don, Rachel, and Susie live in Richmond.Ed, Fred, Terri, and Uzi live in Charlottesville.Greg, Hank, Valerie, and Wendy live in Roanoke.
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Example
Example (Cluster sample)Use cluster sampling to choose a sample of size n = 8,where the clusters are the cities.Is the sample representative with regard to sex?
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Stratified Sampling vs. Cluster Sampling
In stratified samplingFrom all of the strata we take randomly selectedindividuals.
In cluster samplingFrom randomly selected clusters we take all of theindividuals.
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Stratified andCluster
Sampling
Robb T.Koether
HomeworkReview
Introduction
StratifiedRandomSamplesExample
EstimatingParameters
ClusterSamplesExample
Stratified vs.Cluster
Assignment
Assignment
HomeworkRead Sections 2.6, 2.8, pages 108 - 115, 122 - 126.Let’s Do It! 2.6, 2.8.Page 115, exercises 19 - 23, 25.Page 126, exercises 35 - 38.